Optimal. Leaf size=297 \[ \frac{2 a^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{15 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 a^{5/4} B \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{2 \sqrt{e x} \sqrt{a+c x^2} (5 A+3 B x)}{15 e}+\frac{4 a B x \sqrt{a+c x^2}}{5 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]
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Rubi [A] time = 0.271838, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {815, 842, 840, 1198, 220, 1196} \[ \frac{2 a^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (3 \sqrt{a} B+5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 a^{5/4} B \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{2 \sqrt{e x} \sqrt{a+c x^2} (5 A+3 B x)}{15 e}+\frac{4 a B x \sqrt{a+c x^2}}{5 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]
Antiderivative was successfully verified.
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Rule 815
Rule 842
Rule 840
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{a+c x^2}}{\sqrt{e x}} \, dx &=\frac{2 \sqrt{e x} (5 A+3 B x) \sqrt{a+c x^2}}{15 e}+\frac{4 \int \frac{\frac{5}{2} a A c e^2+\frac{3}{2} a B c e^2 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{15 c e^2}\\ &=\frac{2 \sqrt{e x} (5 A+3 B x) \sqrt{a+c x^2}}{15 e}+\frac{\left (4 \sqrt{x}\right ) \int \frac{\frac{5}{2} a A c e^2+\frac{3}{2} a B c e^2 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{15 c e^2 \sqrt{e x}}\\ &=\frac{2 \sqrt{e x} (5 A+3 B x) \sqrt{a+c x^2}}{15 e}+\frac{\left (8 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{\frac{5}{2} a A c e^2+\frac{3}{2} a B c e^2 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{15 c e^2 \sqrt{e x}}\\ &=\frac{2 \sqrt{e x} (5 A+3 B x) \sqrt{a+c x^2}}{15 e}-\frac{\left (4 a^{3/2} B \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{5 \sqrt{c} \sqrt{e x}}+\frac{\left (4 a \left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{15 \sqrt{c} \sqrt{e x}}\\ &=\frac{2 \sqrt{e x} (5 A+3 B x) \sqrt{a+c x^2}}{15 e}+\frac{4 a B x \sqrt{a+c x^2}}{5 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{4 a^{5/4} B \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{2 a^{3/4} \left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0263668, size = 82, normalized size = 0.28 \[ \frac{2 x \sqrt{a+c x^2} \left (3 A \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{a}\right )+B x \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{a}\right )\right )}{3 \sqrt{e x} \sqrt{\frac{c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 312, normalized size = 1.1 \begin{align*}{\frac{2}{15\,c} \left ( 5\,A\sqrt{-ac}\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) a+6\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}-3\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}+3\,B{c}^{2}{x}^{4}+5\,A{c}^{2}{x}^{3}+3\,aBc{x}^{2}+5\,aAcx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + a}{\left (B x + A\right )}}{\sqrt{e x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + a}{\left (B x + A\right )} \sqrt{e x}}{e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.39806, size = 97, normalized size = 0.33 \begin{align*} \frac{A \sqrt{a} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{5}{4}\right )} + \frac{B \sqrt{a} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{7}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + a}{\left (B x + A\right )}}{\sqrt{e x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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